The classical cubic law deviates from the estimation of the flow rate because of the surface roughness distributed along the fracture surface, which can be decomposed as the primary and the secondary roughness. A piecewise equivalent model is developed to characterise the relationship between primary and secondary roughness and flow rate inside the fracture, setting the fractal dimension 2.0 as the piecewise point. The rough-walled fracture influenced by the primary and secondary roughness is quantitatively evaluated based on the wavelet transform and the power spectrum method. A classification criterion is then proposed to accurately capture the secondary roughness by investigating the non-linear flow behavior along rough-walled fractures using the lattice Boltzmann method. The results found that secondary roughness greatly enhances the flow non-linearity when the fracture fractal dimension exceeds 2.0, while secondary roughness has little effect on non-linear flow behavior with the fracture fractal dimension less than 2.0. The developed piecewise equivalent model shows that the non-linear flow pattern is affected by both fracture primary and secondary roughness. The flow variation along fractures with identical fracture apertures and primary roughness is greatly related to the influence of the secondary roughness. The increase of secondary roughness greatly decreases the hydraulic aperture and the permeability. This work provides an improved equivalent model which can be potentially used in field practices to precisely predict the non-linear flow property by emphasising the great impacts of the secondary roughness.

由于沿断口分布的表面粗糙度可以分解为初级和次级粗糙度，因此经典三次定律偏离了流量的估计。以分形维数2.0为分段点，建立了分段等效模型来表征裂缝内初级和次级粗糙度与流量之间的关系。基于小波变换和功率谱法定量评价了一次粗糙度和二次粗糙度对粗糙壁裂缝的影响。然后提出了一种分类标准，通过使用格子玻尔兹曼方法研究沿粗糙壁裂缝的非线性流动行为来准确捕获二次粗糙度。结果发现，当裂缝分形维数超过2.0时，次级粗糙度极大地增强了流动非线性，而当裂缝分形维数小于2.0时，次级粗糙度对非线性流动行为的影响很小。开发的分段等效模型表明，非线性流动模式受裂缝初级和次级粗糙度的影响。裂缝开度和原生粗糙度相同的裂缝沿线的流量变化与二次粗糙度的影响有很大关系。二次粗糙度的增加大大降低了水力孔径和渗透率。这项工作提供了一个改进的等效模型，可以潜在地用于现场实践，通过强调二次粗糙度的巨大影响来精确预测非线性流动特性。